Let be the sum of an infinite geometric series with first term and common ratio where

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

S_{\infty}

be the sum of an infinite geometric series with first term

a

and common ratio

r

where

|r|<1

, and let

S_{n}

be the sum of its first

n

terms. (i) Prove that the remainder

R_{n}=S_{\infty}-S_{n}

equals

\dfrac{a r^{n}}{1-r}

. (ii) Show that the sequence

\{R_{n}\}

itself forms a geometric progression, and find its first term and common ratio. (iii) Explain why

R_{n}\to0

n\to\infty

, and relate this fact to the convergence of the original series.